Question

which graph shows the solution to the system of linear inequalities?
x - 4y ≤ 4
y < x + 1

Explanation:

Step1: Rewrite first inequality to slope-intercept form

Rearrange $x - 4y \leq 4$ to solve for $y$:

$$\begin{align*} -4y &\leq -x + 4 \\ y &\geq \frac{1}{4}x - 1 \end{align*}$$

This is a solid line (due to $\leq$) with slope $\frac{1}{4}$, y-intercept $-1$, and shading above the line.

Step2: Analyze second inequality

$y < x + 1$ is a dashed line (due to $<$) with slope $1$, y-intercept $1$, and shading below the line.

Step3: Match to correct graph

Find the graph with:

1. Solid line $y=\frac{1}{4}x - 1$, shaded above it
2. Dashed line $y=x+1$, shaded below it
3. Overlapping shaded region is the solution.

Answer:

The first graph (leftmost option, with solid lower line, dashed upper line, and overlapping shading in the upper-right region relative to the solid line and lower region relative to the dashed line)